Basic Math Examples

$\sqrt{8 a b} - \sqrt{50 a b c^{2}} + \sqrt{18 a b^{3}}$

Step 1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{8 a b}$ as ${(8 a b)}^{\frac{1}{2}}$ .

${(8 a b)}^{\frac{1}{2}} - \sqrt{50 a b c^{2}} + \sqrt{18 a b^{3}}$

Step 2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{50 a b c^{2}}$ as ${(50 a b c^{2})}^{\frac{1}{2}}$ .

${(8 a b)}^{\frac{1}{2}} - {(50 a b c^{2})}^{\frac{1}{2}} + \sqrt{18 a b^{3}}$

Step 3

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{18 a b^{3}}$ as ${(18 a b^{3})}^{\frac{1}{2}}$ .

${(8 a b)}^{\frac{1}{2}} - {(50 a b c^{2})}^{\frac{1}{2}} + {(18 a b^{3})}^{\frac{1}{2}}$

Step 4

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 4.1

Apply the product rule to $8 a b$ .

${(8 a)}^{\frac{1}{2}} b^{\frac{1}{2}} - {(50 a b c^{2})}^{\frac{1}{2}} + {(18 a b^{3})}^{\frac{1}{2}}$

Step 4.2

Apply the product rule to $8 a$ .

$8^{\frac{1}{2}} a^{\frac{1}{2}} b^{\frac{1}{2}} - {(50 a b c^{2})}^{\frac{1}{2}} + {(18 a b^{3})}^{\frac{1}{2}}$

Step 5

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 5.1

Apply the product rule to $50 a b c^{2}$ .

$8^{\frac{1}{2}} a^{\frac{1}{2}} b^{\frac{1}{2}} - ({(50 a b)}^{\frac{1}{2}} {(c^{2})}^{\frac{1}{2}}) + {(18 a b^{3})}^{\frac{1}{2}}$

Step 5.2

Apply the product rule to $50 a b$ .

$8^{\frac{1}{2}} a^{\frac{1}{2}} b^{\frac{1}{2}} - ({(50 a)}^{\frac{1}{2}} b^{\frac{1}{2}} {(c^{2})}^{\frac{1}{2}}) + {(18 a b^{3})}^{\frac{1}{2}}$

Step 5.3

Apply the product rule to $50 a$ .

$8^{\frac{1}{2}} a^{\frac{1}{2}} b^{\frac{1}{2}} - (50^{\frac{1}{2}} a^{\frac{1}{2}} b^{\frac{1}{2}} {(c^{2})}^{\frac{1}{2}}) + {(18 a b^{3})}^{\frac{1}{2}}$

Step 6

Multiply the exponents in ${(c^{2})}^{\frac{1}{2}}$ .

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Step 6.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$8^{\frac{1}{2}} a^{\frac{1}{2}} b^{\frac{1}{2}} - (50^{\frac{1}{2}} a^{\frac{1}{2}} b^{\frac{1}{2}} c^{2 (\frac{1}{2})}) + {(18 a b^{3})}^{\frac{1}{2}}$

Step 6.2

Cancel the common factor of $2$ .

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Step 6.2.1

Cancel the common factor.

$8^{\frac{1}{2}} a^{\frac{1}{2}} b^{\frac{1}{2}} - (50^{\frac{1}{2}} a^{\frac{1}{2}} b^{\frac{1}{2}} c^{2 (\frac{1}{2})}) + {(18 a b^{3})}^{\frac{1}{2}}$

Step 6.2.2

Rewrite the expression.

$8^{\frac{1}{2}} a^{\frac{1}{2}} b^{\frac{1}{2}} - (50^{\frac{1}{2}} a^{\frac{1}{2}} b^{\frac{1}{2}} c^{1}) + {(18 a b^{3})}^{\frac{1}{2}}$

Step 7

Simplify.

$8^{\frac{1}{2}} a^{\frac{1}{2}} b^{\frac{1}{2}} - (50^{\frac{1}{2}} a^{\frac{1}{2}} b^{\frac{1}{2}} c) + {(18 a b^{3})}^{\frac{1}{2}}$

Step 8

Remove parentheses.

$8^{\frac{1}{2}} a^{\frac{1}{2}} b^{\frac{1}{2}} - (50^{\frac{1}{2}} a^{\frac{1}{2}} b^{\frac{1}{2}}) c + {(18 a b^{3})}^{\frac{1}{2}}$

Step 9

Remove parentheses.

$8^{\frac{1}{2}} a^{\frac{1}{2}} b^{\frac{1}{2}} - (50^{\frac{1}{2}} a^{\frac{1}{2}}) b^{\frac{1}{2}} c + {(18 a b^{3})}^{\frac{1}{2}}$

Step 10

Remove parentheses.

$8^{\frac{1}{2}} a^{\frac{1}{2}} b^{\frac{1}{2}} - 50^{\frac{1}{2}} a^{\frac{1}{2}} b^{\frac{1}{2}} c + {(18 a b^{3})}^{\frac{1}{2}}$

Step 11

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 11.1

Apply the product rule to $18 a b^{3}$ .

$8^{\frac{1}{2}} a^{\frac{1}{2}} b^{\frac{1}{2}} - 50^{\frac{1}{2}} a^{\frac{1}{2}} b^{\frac{1}{2}} c + {(18 a)}^{\frac{1}{2}} {(b^{3})}^{\frac{1}{2}}$

Step 11.2

Apply the product rule to $18 a$ .

$8^{\frac{1}{2}} a^{\frac{1}{2}} b^{\frac{1}{2}} - 50^{\frac{1}{2}} a^{\frac{1}{2}} b^{\frac{1}{2}} c + 18^{\frac{1}{2}} a^{\frac{1}{2}} {(b^{3})}^{\frac{1}{2}}$

Step 12

Multiply the exponents in ${(b^{3})}^{\frac{1}{2}}$ .

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Step 12.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$8^{\frac{1}{2}} a^{\frac{1}{2}} b^{\frac{1}{2}} - 50^{\frac{1}{2}} a^{\frac{1}{2}} b^{\frac{1}{2}} c + 18^{\frac{1}{2}} a^{\frac{1}{2}} b^{3 (\frac{1}{2})}$

Step 12.2

Combine $3$ and $\frac{1}{2}$ .

$8^{\frac{1}{2}} a^{\frac{1}{2}} b^{\frac{1}{2}} - 50^{\frac{1}{2}} a^{\frac{1}{2}} b^{\frac{1}{2}} c + 18^{\frac{1}{2}} a^{\frac{1}{2}} b^{\frac{3}{2}}$

Step 13

Rewrite $8^{\frac{1}{2}} a^{\frac{1}{2}} b^{\frac{1}{2}} - 50^{\frac{1}{2}} a^{\frac{1}{2}} b^{\frac{1}{2}} c + 18^{\frac{1}{2}} a^{\frac{1}{2}} b^{\frac{3}{2}}$ in a factored form.

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Step 13.1

Factor $a^{\frac{1}{2}} b^{\frac{1}{2}}$ out of $8^{\frac{1}{2}} a^{\frac{1}{2}} b^{\frac{1}{2}} - 50^{\frac{1}{2}} a^{\frac{1}{2}} b^{\frac{1}{2}} c + 18^{\frac{1}{2}} a^{\frac{1}{2}} b^{\frac{3}{2}}$ .

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Step 13.1.1

Factor $a^{\frac{1}{2}} b^{\frac{1}{2}}$ out of $8^{\frac{1}{2}} a^{\frac{1}{2}} b^{\frac{1}{2}}$ .

$a^{\frac{1}{2}} b^{\frac{1}{2}} (8^{\frac{1}{2}}) - 50^{\frac{1}{2}} a^{\frac{1}{2}} b^{\frac{1}{2}} c + 18^{\frac{1}{2}} a^{\frac{1}{2}} b^{\frac{3}{2}}$

Step 13.1.2

Factor $a^{\frac{1}{2}} b^{\frac{1}{2}}$ out of $- 50^{\frac{1}{2}} a^{\frac{1}{2}} b^{\frac{1}{2}} c$ .

$a^{\frac{1}{2}} b^{\frac{1}{2}} (8^{\frac{1}{2}}) + a^{\frac{1}{2}} b^{\frac{1}{2}} (- 50^{\frac{1}{2}} c) + 18^{\frac{1}{2}} a^{\frac{1}{2}} b^{\frac{3}{2}}$

Step 13.1.3

Factor $a^{\frac{1}{2}} b^{\frac{1}{2}}$ out of $18^{\frac{1}{2}} a^{\frac{1}{2}} b^{\frac{3}{2}}$ .

$a^{\frac{1}{2}} b^{\frac{1}{2}} (8^{\frac{1}{2}}) + a^{\frac{1}{2}} b^{\frac{1}{2}} (- 50^{\frac{1}{2}} c) + a^{\frac{1}{2}} b^{\frac{1}{2}} (18^{\frac{1}{2}} b^{\frac{2}{2}})$

Step 13.1.4

Factor $a^{\frac{1}{2}} b^{\frac{1}{2}}$ out of $a^{\frac{1}{2}} b^{\frac{1}{2}} (8^{\frac{1}{2}}) + a^{\frac{1}{2}} b^{\frac{1}{2}} (- 50^{\frac{1}{2}} c)$ .

$a^{\frac{1}{2}} b^{\frac{1}{2}} (8^{\frac{1}{2}} - 50^{\frac{1}{2}} c) + a^{\frac{1}{2}} b^{\frac{1}{2}} (18^{\frac{1}{2}} b^{\frac{2}{2}})$

Step 13.1.5

Factor $a^{\frac{1}{2}} b^{\frac{1}{2}}$ out of $a^{\frac{1}{2}} b^{\frac{1}{2}} (8^{\frac{1}{2}} - 50^{\frac{1}{2}} c) + a^{\frac{1}{2}} b^{\frac{1}{2}} (18^{\frac{1}{2}} b^{\frac{2}{2}})$ .

$a^{\frac{1}{2}} b^{\frac{1}{2}} (8^{\frac{1}{2}} - 50^{\frac{1}{2}} c + 18^{\frac{1}{2}} b^{\frac{2}{2}})$

Step 13.2

Divide $2$ by $2$ .

$a^{\frac{1}{2}} b^{\frac{1}{2}} (8^{\frac{1}{2}} - 50^{\frac{1}{2}} c + 18^{\frac{1}{2}} b^{1})$

Step 13.3

Simplify.

$a^{\frac{1}{2}} b^{\frac{1}{2}} (8^{\frac{1}{2}} - 50^{\frac{1}{2}} c + 18^{\frac{1}{2}} b)$

Step 14

Factor.

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Step 14.1

Factor $- 1$ out of $8^{\frac{1}{2}} - 50^{\frac{1}{2}} c + 18^{\frac{1}{2}} b$ .

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Step 14.1.1

Reorder the expression.

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Step 14.1.1.1

Move $8^{\frac{1}{2}}$ .

$a^{\frac{1}{2}} b^{\frac{1}{2}} (- 50^{\frac{1}{2}} c + 18^{\frac{1}{2}} b + 8^{\frac{1}{2}})$

Step 14.1.1.2

Reorder $- 50^{\frac{1}{2}} c$ and $18^{\frac{1}{2}} b$ .

$a^{\frac{1}{2}} b^{\frac{1}{2}} (18^{\frac{1}{2}} b - 50^{\frac{1}{2}} c + 8^{\frac{1}{2}})$

Step 14.1.2

Factor $- 1$ out of $18^{\frac{1}{2}} b$ .

$a^{\frac{1}{2}} b^{\frac{1}{2}} (- (- 18^{\frac{1}{2}} b) - 50^{\frac{1}{2}} c + 8^{\frac{1}{2}})$

Step 14.1.3

Factor $- 1$ out of $- 50^{\frac{1}{2}} c$ .

$a^{\frac{1}{2}} b^{\frac{1}{2}} (- (- 18^{\frac{1}{2}} b) - (50^{\frac{1}{2}} c) + 8^{\frac{1}{2}})$

Step 14.1.4

Factor $- 1$ out of $8^{\frac{1}{2}}$ .

$a^{\frac{1}{2}} b^{\frac{1}{2}} (- (- 18^{\frac{1}{2}} b) - (50^{\frac{1}{2}} c) - 1 (- 8^{\frac{1}{2}}))$

Step 14.1.5

Factor $- 1$ out of $- (- 18^{\frac{1}{2}} b) - (50^{\frac{1}{2}} c)$ .

$a^{\frac{1}{2}} b^{\frac{1}{2}} (- (- 18^{\frac{1}{2}} b + 50^{\frac{1}{2}} c) - 1 (- 8^{\frac{1}{2}}))$

Step 14.1.6

Factor $- 1$ out of $- (- 18^{\frac{1}{2}} b + 50^{\frac{1}{2}} c) - 1 (- 8^{\frac{1}{2}})$ .

$a^{\frac{1}{2}} b^{\frac{1}{2}} (- (- 18^{\frac{1}{2}} b + 50^{\frac{1}{2}} c - 8^{\frac{1}{2}}))$

Step 14.2

Remove unnecessary parentheses.

$a^{\frac{1}{2}} b^{\frac{1}{2}} \cdot - 1 (- 18^{\frac{1}{2}} b + 50^{\frac{1}{2}} c - 8^{\frac{1}{2}})$

Step 15

Factor out negative.

$- (a^{\frac{1}{2}} b^{\frac{1}{2}} (- 18^{\frac{1}{2}} b + 50^{\frac{1}{2}} c - 8^{\frac{1}{2}}))$

Step 16

Remove unnecessary parentheses.

$- a^{\frac{1}{2}} b^{\frac{1}{2}} (- 18^{\frac{1}{2}} b + 50^{\frac{1}{2}} c - 8^{\frac{1}{2}})$

Step 17

Factor.

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Step 17.1

Factor $- 1$ out of $- 18^{\frac{1}{2}} b + 50^{\frac{1}{2}} c - 8^{\frac{1}{2}}$ .

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Step 17.1.1

Factor $- 1$ out of $- 18^{\frac{1}{2}} b$ .

$- a^{\frac{1}{2}} b^{\frac{1}{2}} (- (18^{\frac{1}{2}} b) + 50^{\frac{1}{2}} c - 8^{\frac{1}{2}})$

Step 17.1.2

Factor $- 1$ out of $50^{\frac{1}{2}} c$ .

$- a^{\frac{1}{2}} b^{\frac{1}{2}} (- (18^{\frac{1}{2}} b) - (- 50^{\frac{1}{2}} c) - 8^{\frac{1}{2}})$

Step 17.1.3

Factor $- 1$ out of $- 8^{\frac{1}{2}}$ .

$- a^{\frac{1}{2}} b^{\frac{1}{2}} (- (18^{\frac{1}{2}} b) - (- 50^{\frac{1}{2}} c) - (8^{\frac{1}{2}}))$

Step 17.1.4

Factor $- 1$ out of $- (18^{\frac{1}{2}} b) - (- 50^{\frac{1}{2}} c)$ .

$- a^{\frac{1}{2}} b^{\frac{1}{2}} (- (18^{\frac{1}{2}} b - 50^{\frac{1}{2}} c) - (8^{\frac{1}{2}}))$

Step 17.1.5

Factor $- 1$ out of $- (18^{\frac{1}{2}} b - 50^{\frac{1}{2}} c) - (8^{\frac{1}{2}})$ .

$- a^{\frac{1}{2}} b^{\frac{1}{2}} (- (18^{\frac{1}{2}} b - 50^{\frac{1}{2}} c + 8^{\frac{1}{2}}))$

Step 17.2

Remove unnecessary parentheses.

$- a^{\frac{1}{2}} b^{\frac{1}{2}} \cdot - 1 (18^{\frac{1}{2}} b - 50^{\frac{1}{2}} c + 8^{\frac{1}{2}})$

Step 18

Combine exponents.

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Step 18.1

Factor out negative.

$- (a^{\frac{1}{2}} b^{\frac{1}{2}} \cdot - 1 (18^{\frac{1}{2}} b - 50^{\frac{1}{2}} c + 8^{\frac{1}{2}}))$

Step 18.2

Multiply $- 1$ by $- 1$ .

$1 (a^{\frac{1}{2}} b^{\frac{1}{2}} (18^{\frac{1}{2}} b - 50^{\frac{1}{2}} c + 8^{\frac{1}{2}}))$

Step 18.3

Multiply $a^{\frac{1}{2}}$ by $1$ .

$a^{\frac{1}{2}} (b^{\frac{1}{2}} (18^{\frac{1}{2}} b - 50^{\frac{1}{2}} c + 8^{\frac{1}{2}}))$

Step 19

Remove unnecessary parentheses.

$a^{\frac{1}{2}} b^{\frac{1}{2}} (18^{\frac{1}{2}} b - 50^{\frac{1}{2}} c + 8^{\frac{1}{2}})$